Eureka
The story of Archimedes leaping from his bath and running through Syracuse is true enough — it comes from Vitruvius, written two centuries later. What matters is the principle he found, which is still the correct statement of how things float.
Around the middle of the third century BC, a mathematician named Archimedes was asked by the king of Syracuse to solve a problem that sounded simple: determine whether a crown was made of pure gold without melting it down. The problem was harder than it looked, and the answer Archimedes found — wherever and however it arrived — was one of the first quantitative principles in the history of physics. Two thousand years later it is still taught unchanged.
The man who commissioned the crown was Hiero II, ruler of Syracuse on the southeastern tip of Sicily. He had grown wealthy enough to want a ceremonial gold crown, and wealthy enough to be suspicious. The goldsmith who made it had been given a measured weight of gold. The finished crown matched that weight precisely. But Hiero suspected the man had kept some of the gold and replaced it with silver, which was less valuable and would have been undetectable by a simple weighing — silver being lighter than gold, the alloy could be blended to hit the same mass with a larger volume.
The question was whether the crown was what it was supposed to be. And the constraint was that the king did not want the crown destroyed or even melted, which would have made the test straightforward. He brought the problem to Archimedes.
The version Vitruvius tells
What happened next was recorded by the Roman architect Vitruvius in his De Architectura, written around 25 BC — approximately two centuries after Archimedes' death. Vitruvius is the primary source for the famous story, and the distance in time matters, because it means the account should be read as a well-established tradition rather than a direct record.
According to Vitruvius, Archimedes was still turning the problem over in his mind when he went to a public bath. He climbed into the full basin, and noticed that as he settled into the water, a volume of it overflowed and spilled over the rim. He saw immediately what this implied. He was so seized by the insight that he ran through the streets back to his house without stopping to dress, shouting as he went: Eureka! Eureka! — "I have found it! I have found it!"
The story is irresistible and has been retold ever since. It is also incomplete. Vitruvius reports the discovery but does not give the mathematics. He does not explain how Archimedes applied the principle to the crown — whether by immersing the crown and an equal weight of pure gold separately, or by some other comparison — and historians of science have noted that a simple immersion test would have been experimentally finicky. The differences in volume between a pure-gold crown and a silver-alloyed one of the same mass would have been real but small, and hard to measure precisely with the equipment available. Whether Archimedes solved the crown problem in the way the story implies, or by some other method entirely, is not recorded.
What is recorded — in his own surviving treatise — is the principle itself.
What the principle actually says
The Eureka moment, whatever its precise circumstances, points to a real physical relationship that Archimedes stated with mathematical exactness in his work On Floating Bodies (De corporibus fluitantibus). The principle is usually stated in modern terms as: a body wholly or partially immersed in a fluid experiences an upward force equal to the weight of the fluid it displaces.
This upward force — now called the buoyant force — is what makes objects float or sink. An object denser than the surrounding fluid displaces less weight of fluid than it weighs, so the net force is downward and the object sinks. An object less dense than the fluid displaces fluid that weighs more than the object itself, so the net force is upward and the object floats. An object of exactly the same density as the fluid is in equilibrium and rests suspended.
Applied to the crown problem: a crown adulterated with silver would be less dense than pure gold. When immersed in water, it would displace more water than an equal-mass piece of pure gold, because a given weight of silver has a larger volume than the same weight of gold. The two objects would have the same weight in air but different apparent weights — that is, different buoyant forces pushing up on them — when submerged. The difference in their apparent underwater weights would reveal whether the crown's density matched pure gold.
This is why the principle was useful. It gave a quantitative test for specific gravity — the ratio of a substance's density to water's — using nothing more than two weighings: one in air, one in water.
The treatise
On Floating Bodies survives in two books and is one of the oldest works of mathematical physics in existence. The first book establishes the basic conditions of hydrostatics: how a fluid at rest behaves, what it means for a surface to be in equilibrium, and the principle of buoyancy stated as a formal proposition. The arguments are geometrical in style, in keeping with all of Archimedes' work — he proves his conclusions from stated axioms by deductive steps, the same method Euclid had applied to geometry a generation earlier.
The second book is more remarkable. It is devoted to determining the conditions under which paraboloids — three-dimensional solids with a parabolic cross-section — will float stably in a given orientation. Archimedes works through the cases systematically: depending on the paraboloid's proportions and its density relative to water, it will float upright, tipped at an angle, or invert. He proves the stable configurations for each case. This is not a practical engineering problem in any obvious sense. It is pure mathematical investigation of the behavior of a class of ideal shapes in a fluid, and it is extraordinarily sophisticated for any period, let alone the third century BC.
The life around the work
Archimedes was born around 287 BC in Syracuse. He likely spent time studying in Alexandria, the intellectual centre of the Greek world in his generation, and may have known the mathematicians working there. He returned to Syracuse and spent most of his career there, corresponding with colleagues in Alexandria by letter — several of his treatises survive in the form of letters addressed to named mathematicians, including Dositheus and Eratosthenes.
His output was wide and deep. On the Sphere and Cylinder established that a sphere inscribed in a cylinder has two-thirds the cylinder's volume — Archimedes considered this his finest result and reportedly asked that the figure be engraved on his tomb. Measurement of a Circle approximated π by inscribing and circumscribing regular polygons around a circle. The Sand Reckoner invented a system for expressing very large numbers and used it to calculate how many grains of sand would fill the universe. These are not exercises in applied calculation; they are investigations into the structure of mathematical reality.
His reputation in antiquity extended also to mechanical inventions. He is credited with a screw-based water pump (the Archimedean screw), with systems of pulleys and levers used in the defense of Syracuse, and — in a story as often repeated as the bath story — with the use of mirrors to set Roman ships alight. The machines used during the Second Punic War siege of Syracuse are described by Polybius and Plutarch, though the details of the mirror story are disputed by modern historians.
The siege and the soldier
In 215 BC, Rome began its siege of Syracuse. The city had allied with Carthage after the Roman defeat at Cannae, and Rome could not leave it in enemy hands. The siege lasted two years, and by several ancient accounts — Polybius, Livy, Plutarch — Archimedes' mechanical devices were a significant factor in its prolongation. Catapults calibrated for various ranges, cranes that lifted Roman ships by their prows and dropped them: the accounts are probably embellished but carry a core of truth. The city fell in 212 BC when a section of the wall was taken while the defenders attended a festival.
The Roman commander Marcellus had ordered that Archimedes not be harmed. According to Plutarch, writing in the first century AD, a Roman soldier encountered Archimedes in the street or at his house, working on a mathematical problem drawn in sand. Archimedes told him not to disturb his diagrams. The soldier killed him.
Plutarch's account includes multiple versions of the event, none of them from a source contemporary with Archimedes, and the stories are clearly shaped by the rhetorical purposes of the ancient writers. What they preserve is a tradition: that Archimedes died while working, that he was recognizable enough that Marcellus mourned him, and that his death was treated as a loss worth recording. Cicero, visiting Syracuse around 75 BC, found what he believed to be Archimedes' tomb and confirmed the sphere-and-cylinder carving.
The Eureka story is a simpler and more vivid piece of tradition — a discovery in a bath, a man running through a city, a phrase that two millennia of writers have not been able to leave alone. What it points to, beneath the legend, is a real work: two books of hydrostatics, the oldest quantitative treatment of a physical law that survives. A body immersed in a fluid is pushed up by the weight of fluid it displaces. Archimedes proved it. It is still true.